Efficient Parallel Solutions of Large Sparse Spd Systems on Distributed-memory Multiprocessors

We consider several issues involved in the solution of sparse symmetric positive deenite systems by multifrontal method on distributed-memory multiprocessors. First, we present a new algorithm for computing the partial factorization of a frontal matrix on a subset of processors which signiicantly improves the performance of a distributed multifrontal algorithm previously designed. Second, new parallel algorithms for computing sparse forward elimination and sparse backward substitution are described. The new algorithms solve the sparse triangular systems in a multifrontal fashion. Numerical experiments run on an Intel iPSC/860 and an Intel iPSC/2 for a set of problems with regular and irregular sparsity structure are reported. More than 180 million ops per second during the numerical factorization are achieved for a three-dimensional grid problem on an iPSC/860 machine with 32 processors. 1. Introduction. The numerical solution of a large sparse symmetric positive deenite system Ax = b lies at the heart of many scientiic and engineering computations. One way of solving such a system is rst to compute the sparse Cholesky factorization PAP T = LL T , where L is a lower triangular matrix and P is a permutation matrix for reducing the number of nonzeros in L. The solution vector is computed by solving the two sparse triangular systems Lu = Pb and L T v = u, and setting x = P T v. Typically, a direct solution process involves four steps: (i) computation of a ll-reducing ordering, (ii) symbolic factorization, (iii) numerical factorization, and (iv) triangular solution. George and Liu 13] provide a comprehensive discussion on these four steps and their implementations on serial machines. We focus our attention on the design of eecient parallel algorithms for the numerical factorization and the triangular solution on distributed-memory multiprocessors. We assume that the symbolic processing steps including ordering and symbolic factorization are done sequentially. Signiicant eeort has been invested on designing eecient parallel algorithms for solving sparse symmetric positive deenite systems on distributed-memory multiprocessors. Heath, Ng and Peyton 17] have recently published a survey on parallel sparse matrix factorization algorithms. There are roughly three classes of algorithms for computing sparse Cholesky factorization on distributed-memory machines|fan-out algorithm 12], fan-in algorithm 2] and multifrontal algorithm 8]. Ashcraft, Eisenstat, Liu and Sherman 3] have reported that the multifrontal method is more eecient than both fan-out and fan-in algorithms on an iPSC/2 machine for the 63 63 nine-point regular grid. Pothen and Sun 25] have …